Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor

Let $g$ be a random matrix distributed according to uniform probability measure on the finite general linear group ${\mathrm{GL}}_n\left(F_q\right)$. We show that $\mathrm{Tr}\left(g^k\right)$ equidistributes on $F_q$ as $n\to \infty$ as long as $\log k=o\left(n^2\right)$ and that this range is sharp. We also show that nontrivial linear combinations of $\mathrm{Tr}\left(g^1\right),\text{…},\mathrm{Tr}\left(g^k\right)$ equidistribute as long as $\log k=o\left(n\right)$ and this range is sharp as well. Previously equidistribution of either a single trace or a linear combination of traces was only known for $k⩽c_{q}n$, where $c_q$ depends on $q$, due to work of the first author and Rodgers. We reduce the problem to exhibiting cancellation in certain short character sums in function fields. For the equidistribution of $\mathrm{Tr}\left(g^k\right)$, we end up showing that certain explicit character sums modulo $T^{k+1}$ exhibit cancellation when averaged over monic polynomials of degree $n$ in $F_q\left[T\right]$ as long as $\log k=o\left(n^2\right)$. This goes far beyond the classical range $\log k=o\left(n\right)$ due to Montgomery and Vaughan. To study these sums, we build on the argument of Montgomery and Vaughan but exploit additional symmetry present in the considered sums.